[Math] Particular integral in complementary function

Question

Why does multiplying by $x$ work here? Is this a general rule when the particular integral in complementary function

Answer 1

$$y''-5y'+6y=e^{2x}$$ The characteristic polynomial is: $$r^2-5r+6 =0 \implies (r-2)(r-3)=0$$ $$ \implies S_r= \{2,3\}$$ The solution of the homogeneous equation is : $$y(x)=c_1e^{2x}+c_2e^{3x}$$ So the particular solution should be $$y_p(x)=Axe^{2x}$$ Normally the guess should be $Ae^{2x}$. But since $e^{2x}$ is already solution of the homogeneous equation, you need to multiply by $x$ the guess.

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EDIT

A good exercice is to solve the following equation : $$y''-4y'+4y=e^{2x}$$ Try it ..